Problem 25 - Beginning and Intermediate Algebra

Question

Recently, the movies The Dark Knight, Spider-Man 3, and The Twilight Saga: New Moon held the record for having the three highest-grossing weekends at the box office, with a total of \(\$ 452\) million. Together, Spider-Man 3 and New Moon earned \(\$ 136\) million more than The Dark Knight. New Moon earned \(\$ 15\) million less than The Dark Knight. (Source: the-numbers.com) Find the amount earned by each movie.

Step-by-Step Solution

Answer

In conclusion, the movies earned the following amounts: The Dark Knight earned \(x = \$316\) million, Spider-Man 3 earned \(y = \$135\) million, and New Moon earned \(z = \$301\) million.

Step 1: Set up the equations based on the given information

We can create three equations based on the given information: 1. The total weekend earnings of all three movies: \[x + y + z = 452\] 2. Spider-Man 3 and New Moon together earned $136 million more than The Dark Knight: \[y + z = x + 136\] 3. New Moon earned $15 million less than The Dark Knight: \[z = x - 15\] Now that we have our three equations, we can start solving for the movie earnings.

Step 2: Solve the system of equations for the movie earnings

To solve this system of equations, we can first substitute the third equation into the second equation to eliminate z: \(y + (x - 15) = x + 136 \) Now let's solve for y: \[y = -x + 151\] Next, we can substitute this expression for y back into the first equation to eliminate y: \[x + (-x + 151) + z = 452\] Now we have a single equation containing only x and z: \[151 + z = 452\] Let's solve for z: \[z = 301\] Now that we have found z, we can substitute it back into the third equation to find x: \[301 = x - 15\] \[x = 316\] Now that we have both x and z, we can substitute both of these values back into the expression for y we found in step 2: \[y = -316 + 151\] \[y = 135\]

Step 3: Interpret the results

Finally, we can say that: - The Dark Knight earned: \(x = $316\) million - Spider-Man 3 earned: \(y = $135\) million - New Moon earned: \(z = $301\) million

Key Concepts

Understanding Algebra in Systems of Equations

Algebra is a crucial part of mathematics that allows us to represent real-world scenarios using variables and equations. In the context of solving word problems like the one with movie earnings, algebra helps us form relationships between different unknowns. Instead of dealing with large blocks of numbers, we use variables such as \(x\), \(y\), and \(z\) to represent unknown values.

Algebra allows for logical manipulation of these variables to solve equations. For instance, through algebraic operations, we can express how much more one movie earned compared to another by setting up relationships as equations. This method of solving problems helps simplify even the most complex situations. It all starts with translating words into mathematical equations, effectively breaking down the problem into manageable pieces we can solve step-by-step.

Tackling Word Problems with Equations

Word problems can sometimes seem complicated because they require you to interpret text into mathematical expressions. However, they're just a means of applying mathematical concepts to real-world situations. Let's consider how to tackle a word problem such as determining movie earnings.

When faced with word problems:

• Identify the unknowns. For example, in the problem, we needed to find the earnings of three movies, so we assigned them variables \(x\), \(y\), and \(z\).
• Formulate equations based on given relationships. This involves carefully reading the problem and translating each piece of information into an equation. The statements "Spider-Man 3 and New Moon together earned \$136 million more than The Dark Knight" translates to \(y + z = x + 136\).
• Use these equations to form a system of equations. This is where you bring together all the relationships you've created using algebra.

Writing equations from the narrative forms the backbone of solving word problems effectively. Breaking up the text to isolate each condition separately can make solving these problems much more manageable.

Solving Systems of Linear Equations

A system of equations is a set of multiple equations that are all satisfied by the same set of values. When these equations are linear, it means they graph as straight lines when plotted.

To solve a system of linear equations, especially in real-world scenarios like movie earnings, you typically aim to find the point at which all equations intersect, or in other words, the common solution for all variables.

There are several methods to solve a system:

• Substitution, as used in this exercise, involves solving one equation for a variable and substituting this into another equation. We substituted \(z = x - 15\) into the other equations to find values for \(x\) and \(y\).
• Elimination involves adding or subtracting equations to cancel out a variable, gradually simplifying until solutions are found.
• Graphical methods, though less accurate, involve plotting the equations and finding their intersection graphically.

Understanding these concepts is key for effectively determining the solution to a system of equations, enabling you to resolve many similar algebraic problems efficiently.